S S-DUALITYA nonperturbative symmetry that relates the weak and strong coupling regimes of a quantum theory. In a gauge theory the inversion of the coupling is accompanied by the interchange of the electric and magnetic degrees of freedom [1]. When a h-term is included in the Lagrangian there is an additional symmetry under h ! h þ 2p, and the theory is then invariant under s ! À1=s and s ! s þ 1, withand g the coupling constant. These two transformations generate the SLð2; Z Þ group, or S-duality group [2]. Examples in which it is conjectured to be a symmetry: Four dimensional N¼4 supersymmetric Yang Mills theory, where it is referred as Montonen Olive duality [3]. In string theory: symmetry of the ten dimensional Type IIB superstring theory [4], it relates the Type I and heterotic SO(32) theories [5,6,7], in four dimensions it is a symmetry of the heterotic compactified on a 6-torus [8]. Let G be a real simple Lie algebra with Cartan decomposition G ¼ T È ? P with T a maximal compact subalgebra of G and P the orthogonal complement of T with respect to the Killing form which is negative definite when restricted to T . A linear Cartan involution h is defined by h j P ¼ À1 and h j T ¼ 1. For the split form, the Cartan Chevalley involution h is given by hðh a Þ ¼ Àh a , hðe a Þ ¼ Àe Àa , then, the maximal compact subalgebra T is generated over R by fe a À e Àa g and P by fh a g and fe a þ e Àa g. Let A be a maximal abelian subalgebra of P, and let g be (a maximally non compact h-stable) Cartan subalgebra containing A (A Ã is its dual). The dimension of A corresponds to the real rank of G. For the split form, A is generated over R by the Cartan generators fh a g and the real rank coincides with the rank of its complexification. The conjugation r of G C , the complexification of G, with respect to G defines an involutive automorphism and all the fixed elements of G C by r are the real form G. Thus, we have rðXÞ ¼ X and rðiXÞ ¼ ÀiX 8X 2 G. In the same way, the conjugation s ¼ h:r of K G C is defined with respect to the compact real form G ¼ T È iP. The Satake diagram ðG; AÞ permits to obtain easily the conjugation r and then to find the real form associated to it and consists of: 4. The Dynkin diagram of the root system: R ¼ fa a : a 2 D 1 g.

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